It is important and interesting to determine explicit formulas of the representation number of positive definite quadratic forms.

Here we give the following Lemma, see ( [1] , Theorem 1.64), about the modularity of an eta quotient.

Lemma 1. An eta quotient of level N is a meromorphic modular form of weight on having rational coefficients with respect to q if

a)

b)

c)

For and a nonnegative integer n, we define

Clearly and without loss of generality we can assume that

Now, let’s consider sextenary quadratic forms of the form

where,

We write to denote the number of representations of n by a sextenary quadratic form. Its theta function is obviously

Formulae for for the nine octonary quadratic forms (2i, 2j, 2k, 2l) = (8, 0, 0, 0), (2, 6, 0, 0), (4, 4, 0, 0), (6, 2, 0, 0), (2, 0, 6, 0), (4, 0, 4, 0), (6, 0, 2, 0), (4, 0, 0, 4), and (0, 4, 4, 0) appear in the literature, (cf. [2] - [12] ). Alaca and Williams have obtained some results on sextenary quadratic forms in terms of the functions and, see [13] [14] . There are more works on representation number of sextenary quadratic forms in [15] - [17] . Other methods for representation number have been used in (cf. [7] [10] [12] [18] [19] ). Here, we will classify all fourtuples for which is a modular form of weight 8 with level 24. Then we will obtain their representation numbers in terms of the coefficients of Eisenstein series and some eta quotients.

First, by the following Theorem, we characterize the facts that

are in

Theorem 1. Let

where, , be a sextenary quadratic form. Then its theta series is of the form

Moreover, it is in if and only if is given in the Table 1. Here we also see that are either both even or both odd.

Proof. It follows from the Lemma 1, holomorphicity criterion in ( [20] Corollary 2.3, p. 37) and the fact

7. Kendirli, B. (2012) Cusp Forms in and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic Forms with Discriminant -47. International Journal of Mathematics and Mathematical Sciences, 2012, Article ID: 303492, 10 p.

12. Ramakrishnan B. and Sahu, B. (2013) Evaluation of the Convolution Sums ∑l+15m=nσ(l)σ(m) and ∑3l+5m=n&σ (l) σ(m) and an Application. International Journal of Number Theory, 9, 799-809. http://dx.doi.org/10.1142/S179304211250162X

18. Kendirli, B. (2012) Cusp Forms in S4 (Γ0 (79)) and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic forms With Discriminant -79. Bulletin of the Korean Mathematical Society, 49, 529-572. http://dx.doi.org/10.4134/BKMS.2012.49.3.529

19. Kendirli, B. (2015) The Number of Representations of Positive Integers by Some Direct Sum Of Binary Quadratic Forms With Discriminant -103. IJEMME International Journal of Electronics, Mechanical, and Mecathronics Engineering, 5, 979-1007.